Q -Gorenstein deformation families of Fano varieties or The combinatorics of Mirror Symmetry Alexander Kasprzyk
Fano manifolds Smooth varieties, called manifolds , come with a natural notion of curvature, and fall into one of three classes. Negative curvature Flat Positive curvature General type Calabi–Yau Fano There are finitely many Fano manifolds in each dimension.
Fano manifolds: Basic building blocks of geometry Fano manifolds are the building blocks from which other varieties are formed. Both from the Minimal Model Program And in terms of explicit constructions Fano art by Gemma Anderson
Fano manifolds: Classification The classification of Fano manifolds is known up to dimension 3. Dimension 1 : P 1 (i.e. the Reimann sphere) Dimension 2 (del Pezzo, 1880s): P 2 P 1 × P 1 The blow-up of P 2 in at most 8 points. These are called del Pezzo surfaces. Dimension 3 (Mori–Mukai, 1980s): 105 cases Very little is known in dimension ≥ 4.
Fano polytopes and toric geometry Fix a lattice N ≅ Z n . A convex lattice polytope P ⊂ N ⊗ Q = N Q is Fano if: dim ( P ) = n ; 0 ∈ int ( P ) ; each v ∈ vert ( P ) is a primitive lattice point of N . Two Fano polytopes P and Q are considered to be isomorphic if there exists a change of basis of N sending P to Q . That is, ≅ ⇐ ⇒ ϕ ( P ) = Q , for some ϕ ∈ GL n ( Z ) P Q ϕ ∶ e 1 ↦ e 1 ≅ e 2 ↦ e 1 − e 2 via We consider Fano polytopes only up to isomorphism.
Fano polytopes and toric geometry To a Fano polytope P ⊂ N Q we associate the spanning fan . The spanning fan describes a toric Fano variety X P . X P = P 2 ← → ← → The geometry of X P is encoded in the combinatorics of P . For example, the singularities of X P can be read off P .
Toric Fano manifolds: Classification A Fano polytope P is smooth if: For each facet F of P , vert ( F ) are a Z -basis of N . n -dimensional toric Fano smooth Fano polytope P with dim ( P ) = n manifold X toric geometry ← � � � � � � → Dimension 2 : P 1 × P 1 ; the blow-up of P 2 in at most 3 points. P 2 ;
Toric Fano manifolds: Classification Being toric is unusual : Dimension 2 : 5 of the 10 del Pezzo surfaces are toric. Dimension 3 : 18 of the 105 Fano manifolds are toric. But being toric is good : we can use the combinatorics of lattice polytopes to study them. For example, Øbro (2007) gave an efficient algorithm for classifying smooth Fano polytopes in any dimension. Dimension 1 2 3 4 5 6 7 8 Number 1 5 18 124 866 7 622 72 256 749 892 They grow slowly – approximately by a power of 10 per dimension.
Mirror Symmetry n -dimensional Fano Laurent polynomial f manifold X in n variables ← � � � � � � � � → f = x + y + z + 1 Mirror Symmetry xyz � � Newt ( f ) � � deformation � � � � → → n -dimensional toric Fano polytope P with dim ( P ) = n Fano variety X P ← � � � � � � → toric geometry
Example: P 2 Illustrate this equivalence in the case of X = P 2 . We start with the Laurent polynomial f = x + y + 1 xy ∈ C [ x ± 1 , y ± 1 ] Associated with f is its period π f ( t ) = ( 1 2 t ∈ C , ∣ t ∣ ≪ ∞ . 1 dx dy 2 π i ) ∫ ∣ x ∣=∣ y ∣= 1 1 − tf y , x The Taylor expansion of the period has coefficients given by the constant term of successive powers of f π f ( t ) = ∑ coeff 1 ( f k ) t k = 1 + 6 t 3 + 90 t 6 + 34650 t 9 + 756756 t 12 + 17153136 t 15 + ... k ≥ 0 ( 3 k ) ! = ∑ ( k ! ) 3 t 3 k k ≥ 0
Example: P 2 π f ( t ) = 1 + 6 t 3 + 90 t 6 + 34650 t 9 + 756756 t 12 + 17153136 t 15 + ... The coefficients of π f agree with certain Gromov–Witten invariants of X . Roughly speaking, they count curves in X with given degree and a certain constraint on the C -structure. This is called the regularised quantum period ̂ G X . f is mirror dual to X if π f = ̂ G X The Newton polytope P ⊂ N Q of f gives a toric Fano variety X P Q -Gorenstein deformation equivalent to X . In this case we recover P 2 . f = x + y + 1 P = Newt ( f ) = ⊂ N Q xy
Example: P 2 The mirror f for X is typically not unique. One way of transforming f to a mirror-equivalent Laurent polynomial g is via a mutation . This is a change of variables ϕ ∶ ( C × ) n ⇢ ( C × ) n such that g = ϕ ∗ f is a Laurent polynomial with the same period: π f ( t ) = π g ( t ) In the case f = x + y + 1 xy we can apply the mutation x ↦ x 1 + x ϕ ∶ y y ↦ y 1 + x y Then: ( 1 + x y ) 2 g = ϕ ∗ f = ϕ ∗ ( x + y + 1 xy ) = + + x y 1 + x 1 + x xy y y
Example: P 2 ( 1 + x y ) 2 g = ϕ ∗ f = + + x y 1 + x 1 + x xy y + y 2 + 2 xy + x 2 y = y ( y + x ) y + x xy 3 = y + 1 xy + 2 y 2 + x y 3 ∈ C [ x ± 1 , y ± 1 ] One can compute the period of g : π g ( t ) = 1 + 6 t 3 + 90 t 6 + 34650 t 9 + 756756 t 12 + ⋯ = π f ( t ) g is also a mirror for P 2
Mutation of a Laurent polynomial A mutation of f ∈ C [ x ± 1 ] requires two pieces of data: a grading on monomials; a factor F ∈ C [ x ± 1 ] . The grading is a map w ∶ x a ↦ w ( a ) from monomials to Z . The factor is a Laurent polynomial with w ( F ) = { 0 } such that f h = F − h r h , for all h < 0, where r h ∈ C [ x ± 1 ] . Here f h = “the terms of f in graded piece h ” , i.e. w ( f h ) = { h } . Then ϕ ∶ x a ↦ x a F w ( a ) is a mutation of f with g = ϕ ∗ f = ∑ r h + ∑ f h F h h < 0 h ≥ 0
Example: P 2 Mutation is a combinatorial operation on the Newton polytopes At the level of Newton polytopes we have transformed the Fano polygon for P 2 into the Fano polygon for P ( 1 , 1 , 4 ) : Newt ( x + y + 1 xy ) = � → = Newt ( y + 1 xy + 2 y 2 + x y 3 ) Notice that P ( 1 , 1 , 4 ) is a singular toric Fano variety. It has two smooth 4 ( 1 , 1 ) singularity. cones, and one singular cone corresponding to a 1
Mutation of P ⊂ N Q A mutation of P ⊂ N Q requires two pieces of data: a grading on N ; a factor of P . The grading is given by a primitive lattice vector w ∈ M = Hom ( N , Z ) . The factor is a convex lattice polytope F ⊂ w ⊥ ⊂ N Q such that { v ∈ vert ( P ) ∣ w ( v ) = h } ⊂ ( − h ) F + R h ⊂ P h , for all h < 0, where R h ⊂ N Q is a convex lattice polytope. Here P h = conv ( v ∈ P ∩ N ∣ w ( v ) = h ) . The the mutation of P is Q = conv ( ⋃ R h ∪ ⋃ ( P h + hF )) h < 0 h ≥ 0
Mutation of P ⊂ N Q In the example of P 2 we pick F = conv {( 0 , 0 ) , ( 1 , − 1 )} ⊂ w ⊥ ⊂ N Q . w = (− 1 , − 1 ) ∈ M , Then mutation adds or subtracts dilates of F depending on height: � → -2 -1 -2 0 1 -1 2 0 1 2
Example: P 2 Now consider the dual polytope to P ⊂ N Q : P ∗ = { u ∈ M Q ∣ u ( v ) ≥ − 1 for all v ∈ P } N Q M Q P 2 ∶ � → dual � → P ( 1 , 1 , 4 ) ∶ � → dual
Mutation of P ∗ ⊂ M Q Mutation acts via a piecewise GL n ( Z ) map on M : u � → u − w min { w ( v ) ∣ v ∈ vert ( F )} N Q M Q P 2 ∶ � → ( 2 0 ) ( 1 1 ) 1 0 dual − 1 0 � � → → P ( 1 , 1 , 4 ) ∶ � → dual
Mutation of P ∗ ⊂ M Q P ∗ = � → = Q ∗ Mutation has straightened out the bottom-left corner of Q ∗ . Since this is a piecewise GL n ( Z ) map on M , we have that: Vol ( P ∗ ) = Vol ( Q ∗ ) , Ehr ( P ∗ ) = Ehr ( Q ∗ ) Equivalently: (− K X P ) n = (− K X Q ) n , Hilb ( X P , − K X P ) = Hilb ( X Q , − K X Q )
Mutation of Markov triples We can continue mutating P 2 , moving from Fano triangle to Fano triangle: ( 1 , 1 , 1 ) ( 1 , 1 , 2 ) ( 1 , 2 , 5 ) ( 2 , 5 , 29 ) ( 1 , 5 , 13 ) ( 5 , 29 , 433 ) ( 2 , 29 , 169 ) ( 5 , 13 , 194 ) ( 1 , 13 , 34 ) The vertices ( a , b , c ) correspond to the Fano triangles for P ( a 2 , b 2 , c 2 ) . The vertices ( a , b , c ) correspond to solutions to the Markov equation : a 2 + b 2 + c 2 = 3 abc
Mutation of Markov triples A solution ( a , b , c ) ∈ Z 3 > 0 of the Markov equation a 2 + b 2 + c 2 = 3 abc is called a Markov triple . All Markov triples can be obtained from ( 1 , 1 , 1 ) via mutation : ( a , b , c ) � → ( 3 bc − a , b , c ) Mutations of the Markov triples correspond to mutations of the Fano triangles arising from P 2 .
Quiver mutation We can associate a quiver Q P to a Fano polygon P ⊂ N Q . We have a vertex v i for each edge E i of P . Let w i ∈ M be the primitive (inner) normal vector to E i . Then the number of arrows between vertices v i and v j is given by w i ∧ w j = det ( w i w j ) , where the sign determines the orientation. For P 2 we get: v w 1 = (− 1 , − 1 ) 2 P = Q P = 3 w 2 = (− 1 , 2 ) 3 w 3 = ( 2 , − 1 ) v 1 v 3 3
Quiver mutation We can mutate Q P about a vertex v i . For every path v j � → v i � → v k add in a new edge v j � → v k ; Reverse the direction of every arrow that starts or ends at v i ; Cancel opposing edges. We recover the quiver for P ( 1 , 1 , 4 ) : v 2 P 2 ∶ 3 � → 3 v 1 v 3 3 � � → → v 2 3 P ( 1 , 1 , 4 ) ∶ � → 6 v 1 v 3 3
Mirrors for P 2 v v 2 2 P 2 = 3 3 ∶ � → ∶ = P ( 1 , 1 , 4 ) 6 3 v v 1 1 v v 3 3 3 3 Notice that the quiver for P ( 1 , 1 , 4 ) isn’t balanced . We re-balance by adding multiplicities for to vertices v i given by the edge lengths E i of the Fano polygon. v (1) (1) v 2 2 3 3 Q ( 1 , 1 , 1 ) ∶ � → ∶ Q ( 1 , 1 , 2 ) 6 3 (2) (1) v v 1 v 1 v 3 3 3 3 (1) (1) This re-balancing condition is the Markov equation.
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